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In this note we study graphs $G_r$ with the property that every colouring of $E(G_r)$ with $r+1$ colours admits a copy of some graph $H$ using at most $r$ colours. For $1le rle e(H)$ such graphs occur naturally at intermediate steps in the synthesis of a $2$-colour Ramsey graph $G_1longrightarrow H$. (The corresponding notion of Ramsey-type numbers was introduced by Erdos, Hajnal and Rado in 1965 and subsequently studied by Erdos and Szemeredi in 1972). For $H=K_n$ we prove a result on building a $G_{r}$ from a $G_{r+1}$ and establish Ramsey-infiniteness. From the structural point of view, we characterise the class of the minimal $G_r$ in the case when $H$ is relaxed to be the graph property of containing a cycle; we then use it to progress towards a constructive description of that class by proving both a reduction and an extension theorem.
Building on previous work of the author, for each finite triangle-free graph $mathbf{G}$, we determine the equivalence relation on the copies of $mathbf{G}$ inside the universal homogeneous triangle-free graph, $mathcal{H}_3$, with the smallest numbe
For a $k$-vertex graph $F$ and an $n$-vertex graph $G$, an $F$-tiling in $G$ is a collection of vertex-disjoint copies of $F$ in $G$. For $rin mathbb{N}$, the $r$-independence number of $G$, denoted $alpha_r(G)$ is the largest size of a $K_r$-free se
Analogues of Ramseys Theorem for infinite structures such as the rationals or the Rado graph have been known for some time. In this context, one looks for optimal bounds, called degrees, for the number of colors in an isomorphic substructure rather t
We explore the properties of non-piecewise syndetic sets with positive upper density, which we call discordant, in countable amenable (semi)groups. Sets of this kind are involved in many questions of Ramsey theory and manifest the difference in compl
Combining two classical notions in extremal combinatorics, the study of Ramsey-Turan theory seeks to determine, for integers $mle n$ and $p leq q$, the number $mathsf{RT}_p(n,K_q,m)$, which is the maximum size of an $n$-vertex $K_q$-free graph in whi